I have taken half a dozen big game animals with handguns now and FWIW here is my a$$*le.... er, I mean opinion.
A handgun bullet has a lot more in common with an arrow than it does with a rifle bullet. You are poking a hole in vital organs to cause blood loss, thereby killing the animal. This is in contrast to the hydrostatic shock that will cause an animal hit with a high powered rifle to fall over dead almost instantly. You need the handgun bullet to carry enough energy to reliably penentrate far enough to cause this injury. I don't remember where I picked this up, but a long time ago someone told me that to shoot an animal with a handgun ethically, at the moment of impact it should still have double the energy (in ft/lbs) of the weight of the animal. I have treated this factoid as gossip, and I have yet to make a kill that I didn't consider clean and ethical.
Ballistic coefficient info is avail. on all bullet manufacture's websites, if you are shooting factory ammo so is rough velocity (for handloading, find someone with a chrono). With these two peices of info and the average weight of whatever you are shooting at (cow elk, buck deer, etc), you can calculate your maximum distance to target, and it will amaze you how close you really need to be. I took my first couple of handgun-shot animals with a 44mag, and then upgraded to a 454 and finally to a 460 xvr. This is not to say that you can't kill the heck out of something with a 44mag, just be very careful with your yardages.
Also, forget the JHPs for anything other than humans. Think about it; a 44 caliber solid is larger than an expanded 30cal bullet. Take the reliable penetration (and better BC) of a solid.
I also think long and hard about where the animal can go after the hit. So far, I have been really lucky in that every animal that I have shot with a sidearm has been dead within a matter of seconds or minutes, but you have to think like a bowhunter and not shoot at anything that is really close to a property boundry or an impenetrable (to humans) forest.
On kind of a cool side note, with the load that I developed with my 460xvr, I can ethically shoot a deer well over 150 yards (as far as energy is concerned) so I practice all the time on paper plates out to this distance, and set 150 as my maximum. Last year I hit my buck at 128 yards (prone with a bipod), he took about ten steps and 30 seconds to fall over dead.
Happy hunting! There is nothing like shooting something that big with a handgun.